关于Carpi和Alessandro猜想
On Carpi and Alessandro conjecture
摘要：
著名的公开的Černandi猜想指出，每个具有$n$态的\san都有一个长度至多为$(n-1)^2$的\sw。另一方面，最著名的上限是$n$的三次。最近，在Alessandro和Carpi的论文\cite{CARPI1}中，作者引入了自动机强传递性的新概念，并证明了借助于\emph{Extension}方法，这个性质允许得到最短\sws长度的二次上界.他们也证实了循环自动机的这个猜想。我们证明了这个猜想和长期存在的\emph{Extension}猜想。我们还考虑了广泛使用的扩展方法及其观点。
英文摘要：
The well known open Černý conjecture states that each \san with $n$ states has a \sw of length at most $(n-1)^2$. On the other hand, the best known upper bound is cubic of $n$. Recently, in the paper \cite{CARPI1} of Alessandro and Carpi, the authors introduced the new notion of strongly transitivity for automata and conjectured that this property with a help of \emph{Extension} method allows to get a quadratic upper bound for the length of the shortest \sws. They also confirmed this conjecture for circular automata. We disprove this conjecture and the long-standing \emph{Extension} conjecture too. We also consider the widely used Extension method and its perspectives.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
